Báo cáo toán học: "Largest minimal percolating sets in hypercubes under 2-bootstrap percolation" pdf

Largest minimal percolating sets in hypercubes under 2-bootstrap percolation Eric Riedl University of Notre Dame Department of Mathematics ebriedl@gmail.com Submitted: Oct 10, 2009; Accepted: May 18, 2010; Published: May 25, 2010 Mathematics Subject Classification: 05D99 Abstract Consider the following process, known as r-b ootstrap percolation, on a graph G. Designate some initial infected set A and infect any vertex with at least r infected neighbors, continuing until no new vertices can b e infected. We say A percolates if it eventually infects the entire graph. We s ay A is a minimal percolating set if A percolates, bu t no proper subset percolates. We compute the size of a largest minimal percolating set for r = 2 in the n-dimensional hypercube. 1 Introduction In this paper, we consider the following process, known as r-bootstrap percolation. Desig- nate an initial set A of infected vertices. Let A 0 = A. Then let A t be the set of vertices in A t−1 union the set of vertices which have at least r neighbors in A t−1 . Set A = ∪ i A i , and call  A the set of vertices infected by A. A set A percolates if it infects the entire graph. A percolating set A is said to be minimal if for all v ∈ A the set A \ v does not percolate. Let E(G, r) be the largest size of a minimal percolating set and let m(G, r) be the smallest size of a (necessarily minimal) percolating set. In this paper, we find E(Q n , 2), where Q n is the n-dimensional hypercube and we use similar techniques to find bounds on E([n] d , 2) for all n and d. Since r = 2 for most of this paper, we write E(G) for E(G, 2) without ambiguity. Bootstrap percolation was introduced in 1979 by Chalupa, Leath, and Reich [9] for its applications to dilute magnetic sytems. For more information on the many physical applications of bootstrap percolation, see the survey article by Adler and Lev [1]. Arising naturally from the physical context is the following probabilistic problem. Let each vertex of G be initially infected independently with probability p. Then what is the probability that such a set percolates as a function of p? In part icular, if A is a ra ndomly chosen the electronic journal of combinatorics 17 (2010), #R80 1 set, what is p c (G, r) = inf{p | P(A percolates)  1/2}? Much work has been done on this question. Aizenmann and Lebowitz [2] and Cerf and Cirillo [7] did foundational work towards computing p c ([n] d , r) where [n] d is the n × · · · × n d-dimensional grid. Cerf and Manzo [8] proved that p c ([n] d , r) = Θ  1 log (r−1) n  d−r+1 , where log (r) (x) is log(log(· · · log(x))) (r times). More precise asymptotics were found by Holroyd for r = 2, d = 2 [10] and Balo gh, Bollob´as, Duminil-Copin and Morris [4, 5] for general r and d. Balogh, Peres and Pete [6] determined p c for infinite trees and relate it to the branching order. Considerably less work has been done on finding m(G, r) and E(G, r). For r  d, it is known that n r−1  m([n] d , r)  d r−1 r! n r−1 , where the lower bound follows by Pete [12] and the upp er bound by the method of Schonmann [14]. Ballogh and Bollob´as [3] prove that m([n] d , 2) = ⌈d( n − 1)/2⌉. Pete [12] finds an exact asymptotic for m([n] d , r) when r = d. Morris [11] shows that 4n 2 33  E([n] 2 , 2)  n 2 6 asymptotically, making progress on a question posed by Bollob´as. In [13], an algo r ithm is presented for finding m(T, r) and E(T, r) for all finite trees T , and it is shown that if T is a finite tree with ℓ leaves, m(T, r)  (r−1)|T |+1 r , E(T, r)  r|T |+(r−1)ℓ r+1 and E(T,r)−m(T,r) |T | < r−1 r 2 . In this pap er, we find E( Q n , 2) exactly, and show it to be o n the order of 2 n/4 . First we set some nota tio n. We can represent the vertices of Q n as strings of 0s and 1s of length n, with adjacent vertices being precisely those vertices which differ from each other in exactly one coordinate. We can also represent the vertices of the hypercube as the possible subsets of the set {1, , n}. Recall that the automorphisms of the hypercube are all combinations of the n! permutations of the dimensions and the 2 n reflections. We say that two subsets of the hypercube are isomorphic if there is an automorphism of the hypercube that takes one of them to the other. In this paper, we prove the following main result. Theorem 1. Let 1  r  4 be such that n ≡ r mod 4. Then E(Q n , 2) =      n + 1 0  n  1 n 2  n  10 (1 + 2 r−4 )2 ⌊ n+3 4 ⌋ n  11 Note that E(Q n ,2) E(Q n−1 ,2) does not converge as n → ∞, as it simply cycles around between four different values for large n. For the case of grids, we modify our techniques to obtain the following result. Corollary 2. We have E([n] d , 2)   n 2  d  2 +  j 3  2 ⌊ d−1 3 ⌋ where d ≡ j mod 3, 1  j  3. the electronic journal of combinatorics 17 (2010), #R80 2 x y z Figure 1: The subcube ∗0 1. Note that combining Corollary 2 with a result coming from the proof of Theorem 14 of [11] we obtain  1 4  d  E([n] d , 2)   1 2  2d/3 n d . Note that this shows that E( [n] d , 2) = o(n d ) if and o nly if d = d(n) → ∞ as n → ∞. In Section 2 we review some basic facts about 2-neighbor bootstrap percolation o n [n] d . In Section 3 we describe a construction that is optimal in small dimensions and give a recursive upper bound for E(Q n , 2). In Section 4 we describe a construction tha t is optimal for higher dimensions, and prove optimality by classifying all of the isomorphism classes of largest minimal percolating sets. This gives our main result. In Section 5 we show E([n] d , 2) = O(2 n/3 ) for all fixed n and E(AQ n , 2) = 2 for the augmented hypercube AQ n . 2 Basic facts about 2-percol ation in hypercubes Before proceding, we summarize some basic definitions and facts about 2-percolation in Q n and more generally, grids P n 1 × · · · × P n d . The goal of t his section is to obtain a description of the percolation process in terms of combining subcubes. The material in this section was proven by Balogh and Bollob´as [3]. We say a set S is closed under percolation if S = S. We call a subgraph G of a grid a subgrid if G is itself a grid. We call a subgraph G of a grid or hypercube a subcube if G is a hypercube. Proposition 3. The only subsets of the grid which are closed under percolation are those which are a union of disjoint subgrids that are distance at least three from each other. In the case of hypercubes, the only subgrids of Q n are sub-hypercubes. We represent sub cubes of Q n as strings of 0’s, 1’s and ∗’s, where an ∗ in position i means that the sub cube contains vertices with both 0 and 1 in that position. In particular, the number of ∗’s is the dimension of the subcube. See Figure 1 for an example. We define the kth coordinate of the subcube to be the kth element of the string. Proposition 4. Let A and B be two subgrids of distance at most 2 from each other in a grid G. Then A ∪ B is the smallest subgrid containing both A and B. Moreover, in the case G = Q n if A has coordinates a 1 , , a n and B has coordinates b 1 , , b n where a i , b i ∈ {0, 1, ∗}, then the coordinates of A ∪ B are a i ∨ b i , where x ∨ x = x and x ∨ y = ∗ if x = y. the electronic journal of combinatorics 17 (2010), #R80 3 Figure 2: Two different minimal percolating sets of size 3 in dimension 3. Definition 5. Let A be given, and write A = ∪ i C i , where each C i is a set containing a single point, which is just a 0 -dimensional subcube. Set A 0 = ∪ i {C i }. Then choose a sequence of sets of subgrids A 1 , A 2 , , A k so that A t is identical to A t−1 except that two subgrids B, C ∈ A t−1 within distance 2 of each other are replaced by the subgrid B ∪ C. Require A k to consist of a set of subgrids all of which are distance at least 3 from each other. Then A 0 , · · · , A k is called an execution path of the percolation process. For any execution path, we know that A k = {A}, so A k is independent of execution path. We say a subset S of G is internally spanned by A if A ∩ S = S. Then each B ∈ A i is internally spanned by the vertices that contributed to B in the execution path. Note that the two subgrids B and C which we combine at each step are not necessarily disjoint. Proposition 6. Any percolating set of size at least 2 in Q n will disjointly internally span two subcubes which together span the entire hypercube. Proof. Choose an execution path and take the two hypercubes in A k−1 . 3 An initial construc tion and an up per boun d In this section, we give a simple lower bound that is sharp in low dimensions and a recursive upper bound, which is the key to our entire argument. First we give a simple construction to give an easy lower bound for E(Q n ). Proposition 7. Let A = {10 0 0, 010 0, 001 0, , 000 01}. Then A is a minimal percolating set of size n for n  2. Thus, E(Q n )  n. Proof. The set A clearly p ercolates, and if v i is the vertex with a 1 in the ith coordinate, then A \ v i  will be the Q n−1 with a 0 in the ith coordinate. Note that for A given as in Proposition 7, A \v will simply be a Q n−1 containing the empty set. Moreover, by changing v, we can make A \v range over all such Q n−1 . Thus, for all v, A \ v is as large as it can be given that A is a minimal percolating set. It turns out that this property will complicate our goal of finding E(Q n ) for higher dimensions and that this construction does not give the largest possible E(Q n ) for these dimensions. However, it also will turn out that for 2  n  10 this construction is optimal. See Section 4 for more details. Before proving our recursive upper bound, we need a simple lemma. It is the analog of Lemma 7 of [1 1]. the electronic journal of combinatorics 17 (2010), #R80 4 Lemma 8. We have E(Q n )  E(Q n−1 ). Proof. For n = 1, t he statement is obvious since E(Q 0 ) = 1 a nd E(Q 1 ) = 2. Now suppose n  2, and let A be a minimal percolating set in Q n−1 of largest possible size. We shall construct a minimal percolating set in Q n of size at least |A|. Let P be a fixed sub-Q n−1 contained in Q n , and view A as a subset of P ⊂ Q n . Now, select a vertex v ∈ A. Let w be the unique neighbor of v which does not lie in P. Then A ∪ w percolates in Q n . We claim that A ∪ w \ u does not percolate for every u ∈ A with u = v. This will complete the proof, since it will show that either A ∪ w is a minimal percolating set, or A ∪ w \ v is a minimal percolating set. By minimality of A, we know that B = A \ u will be a union of subcubes of distance at least 3 from each other and will have P \ B nonempty. Since v ∈ B, B ∪ w ∩ P = B, so A \ u does not percolate. This concludes the proof. We now turn our attention to proving a recursive upper bound. The general argument is very similar to an argument found in the proof of the upper bound of Theorem 11 in [3]. However, we include it here because we extract extra information from the proof. The proof relies heavily on the idea of viewing percolatio n as combining nearby subcubes, and it looks at the ways that the last two cubes in the process can b e combined. Proposition 9. We have E(Q n )  max {E(Q n−1 ) + 1, 2E(Q n−4 )}. Proof. Let A ⊂ Q n be a minimal percolating set of size E(Q n ). Since A p ercolates, we know that in any execution path, the final term A k will contain only Q n itself. Hence, the penultimate term, A k−1 will always consist of exactly two subcubes, say P and R, which together infect Q n . Without loss of generality, let dim P  dim R. Now, dim P  n − 1 by minimality of A. Among all execution paths, choose one which has dim P as large as possible. We divide into cases depending on dim P . Case 1 dim P = n − 1. Then there must be a vertex of A outside of P, and that vertex plus A ∩ P will percolate, so R is simply a single point by minimality of A. Hence A is the union o f one vertex and a set which minimally internally spans P , so E(G) = |A|  E(Q n−1 ) + 1 in this case. Case 2 dim P = n − 2. Then we know that there cannot be a vertex of A ∪ R in {v ∈ Q n | d(v, P )  1}, since otherwise we could extend P to a cube of dimension n − 1. Thus, there must be a vertex v of A which has distance 2 from P (since every vertex has distance at most 2 from P) and P ∩ v = Q n (just write out the coordinates of P and v in the 0, 1, ∗ notation). Hence, |A|  E(Q n−2 ) + 1 in this case. Case 3 dim P = n − 3. Then we know that there cannot be a vertex of A ∩ R within distance 2 of P , as this would contradict maximality of dim P . Hence, A ∩ R is contained in a subcube of Q n of distance 3 from P , as the set of vertices which are distance 3 from P is a subcube of dimension n − 3. To see this, note that if, for the electronic journal of combinatorics 17 (2010), #R80 5 example, P = 000∗ ∗, then the set of vertices of distance 3 from P is just 111∗ ∗. Thus, R is contained in this subcube, so d(P, R) = 3, which contradicts the fact that A percolates. Hence, this case cannot occur. Case 4 dim P  n − 4. Then by choice of R, we have dim R  n − 4 as well. Now, P and R are both minimally internally spanned, so |A ∩ P | and |A ∩ R| are each at most E(Q n−4 ). Hence, |A|  2E(Q n−4 ) in this case. In fact, we can get more information from the proof, which we summarize in the following corollary. Corollary 10. If E(Q n ) > E(Q n−1 ) + 1, then any minimal percolating set A of size E(Q n ) has the form A = A 1 ∪ A 2 where A 1 and A 2 are both minimal percolating sets in subcubes of dimension at most n − 4. This result gives a two other nice corollaries. The first is an order of growth upper bound on E. Corollary 11. We have E(Q n ) = O(2 n/4 ). The other is an exact calculation of E for small n. Corollary 12. We have E(Q 0 ) = 1, E(Q 1 ) = 2, and E(Q n ) = n for 2  n  8. Proof. When n  2 the result is easy. For n  3, recall that by Proposition 7 we have E(Q n )  n, so it remains to show E(Q n )  n. For n = 3 the result follows from Corollary 10, as it is not possible to have a subcube of dimension n − 4. For 4  n  8, the result follows from Proposition 9, since 2E(Q n−4 )  n for these n. 4 Jagged sets Now, in light of Proposition 9, given n > 8, we wish to find minimal percolating sets of Q n−4 which we can use to create a minimal percolating set of twice the size in dimension n. The construction from Proposition 7 is unsuitable for this. Proposition 13. Suppose A is a minimal percolating set in Q n with A = B ∪ C the disjoint union of two minimal percolating sets in subcubes of dimension n − 4. Then neither B nor C is isomorphic to the constructioin in Proposition 7. Proof. To see this, suppose we created a percolating set A ⊂ Q n which is a union of one copy of our initial construction B in dimension n − 4 and some minimal percolating set C in a Q n−4 , embedded into two different subcubes of Q n of distance at most 2 from each other. Then in some execution path of the percolation process of A, the penultimate step will consist of precisely these two Q n−4 ’s. To construct such an execution path, simply the electronic journal of combinatorics 17 (2010), #R80 6 combine subcubes in B with subcubes in B and subcubes in C with subcubes in C until B and C are the only two subcubes in A i for some i. Now, in our 0, 1, ∗ nota tion, each Q n−4 will have exactly n − 4 ∗’s. Since n > 8, there must be at least one coor dinate k in which both subcubes have ∗’s. Now, remove the (unique) vertex v from B so that the sub-Q n−5 B \ v  does not have an ∗ in the kth coordinate (we know such a vertex exists from the proof of Proposition 7). Then B \ v and C will still have distance at most 2 from each other, and will still span Q n , so B ∪ C \ v will percolate. Thus, B ∪ C is not minimal. Hence, we cannot use our initial construction to create minimal percolating sets of size n − 4 + E(Q n−4 ) in dimension n. As the above proposition shows, our initial construction is not suitable for constructing large minimal percolating sets in high dimensions. Thus, we define a type of minimal percolating set which is suited to constructing large minimal percolating sets in high dimensions. It is analogous to Morr is’ [11] corner-avoiding minimal percolating sets. Definition 14. We say that a minimal percolating set A in Q n is jagged if for all v ∈ A, A \ v is disjoint from the (n − 2)-dimensional subcube ∗ ∗ 00. Let E ′ (Q n ) be the size of the largest jagged set in Q n . Obviously E ′ (Q n )  E(Q n ). In the following lemma, we use jagged sets to construct large minimal percolating sets in higher dimensions. Lemma 15. We have E ′ (Q n )  2E ′ (Q n−4 ) for n > 5. Proof. Suppose we have a minimal percolating set A in Q n−4 which is jagged. Then we construct a jagged minimal percolating set B of size 2|A| in Q n . We build up our minimal percolating set B in two halves, B 1 and B 2 . For B 1 , we choose a jag ged minimal percolating set isomorphic to A from the subcube ∗ ∗ ∗ ∗ 0001. For B 2 , we choose a j agged minimal percolating set isomorphic to A from the subcube ∗ ∗ 00 ∗ ∗10. Now, this set clearly percolates, as the two subcubes shown span all of Q n and are distance two from each other. We claim that B is minimal. To see this, suppose we remove a vertex v. By swapping coordinates n − 3 and n − 4 with coordinates n − 5 and n − 6 respectively, we can assume without loss of generality that v is from B 1 . Now, since A is jagged, B 1 \ v will be a union of subcubes of distance at least three from each other which have at least one 1 in the (n − 5)-th and (n − 4)-th coordinates. Then each subcube will have distance at least 3 from the others and from B 2 . Hence, B \ v does not percolate, so B is minimal. Moreover, B is jagged because every vertex of B \ v will have either 01 or 10 in the last two coordinates. Now, our construction from Proposition 7 is not jagged for n > 2, as 0 0 is always infected by A \ v for any v. However, we demonstrate a jagged minimal percolating set that uses a lmost as many vertices. the electronic journal of combinatorics 17 (2010), #R80 7 Lemma 16. There exists a jagged percolating set of size n − 1 in dimension n for n  4. Thus, E ′ (Q n )  n − 1. Proof. Let the first n − 2 vertices of A be {v i | 1  i  n −2} where v i is the vertex with a 1 in the ith position and the nth position and 0’s everywhere else. Let the (n−1)st vertex of A be 1 . . . 10. For example, when n = 5, we have A = {10001, 01001, 00101, 1111 0}. The set clearly percolates. The last vertex is obviously necessary for p ercolation, as it is the only vertex without a 01 in the last two coordinates. Now, supp ose we omit one of the other vertices. By permuting the first n − 2 coordinates, we can assume that we omit v 1 . Then all the vertices except t he last combine to form 0 ∗ ∗ ∗ 01 which has distance 3 from 11 1 11 0, so the set does not percolate. Moreover, the set breaks up into two subcubes, each of which has either a 01 or a 10 in the last two coordinates, so it is jagged. Corollary 17. We have E(Q n ) = Θ(2 n/4 ). Proof. By Proposition 9 we know E(Q n ) = O(2 n/4 ) and by Lemma 16 and Lemma 15 we know E(Q n ) = Ω(2 n/4 ). The result follows. In light of Lemma 15, we see that in order to find E(Q n ) for large n, we need only find four sufficiently large consecutive integers with E(Q n ) = E ′ (Q n ). Corollary 18. Suppose there exist four consecutive integers j, · · · , j+3 such that E(Q n ) = E ′ (Q n ) > E(Q n−1 ) for n ∈ {j, · · · , j + 3}. Then for all n > j + 3, E(Q n ) = E ′ (Q n ) = 2E(Q n−4 ). Because of the above Corollary, we need only deal with finitely many cases. We will show that E(Q n ) and E ′ (Q n ) have the values as given in the following chart. This will complete the proof of Theorem 1. n E(Q n ) E ′ (Q n ) 2 2 2 3 3 3 4 4 4 5 5 4 6 6 5 7 7 6 8 8 8 9 9 9 10 10 10 11 12 12 Lemma 19. The values for E(Q n ) and E ′ (Q n ) are as given in the chart. In particular, for 8  n  11, we have E(Q n ) = E ′ (Q n ) = 8 + ⌊2 n−9 ⌋. the electronic journal of combinatorics 17 (2010), #R80 8 Figure 3: The only jagged minimal percolating set of size 4 in dimension 4. Proof. We have assembled most of the ingredients of this proof already. The one major piece that we lack is a classification of minimal percolating sets of size n in dimensions 3  n  7, so we start with this. In dimension 3, there are two isomorphism classes of minimal percolating set, one isomorphic to our initial construction and one jagged. By Corollary 10 we know that any minimal percolating set of size 4 in Q 4 must consist of one vertex plus a minimal percolating set in a sub-Q 3 . Thus, checking case-by-case, we find that there are two isomorphism classes of minimal percolating set in dimension 4, one (the one containing sets isomorphic to {0001, 0011, 1110, 1111}) that is jagged, and another (the one containing sets isomorphic to the one given in Proposition 7) that is not. Similar case-by-case checking shows that in dimension 5, the only minimal percolating sets of size 5 are isomorphic to our initial construction in Proposition 7. Using this, it is not hard to show t hat for n ∈ {6, 7} it also holds that the only minimal percolating sets of size n in Q n are isomorphic to our initial construction. This, together with Corollary 12 and Lemma 16 give the values of E(Q n ) and E ′ (Q n ) as shown in the chart for n  7. We now use the above classification to show that E(Q n ) is at most the value in the table f or 8  n  11. Corollary 12 tells us that E(Q 8 )  8 (and is indeed equal to 8). For 9  n  11, we know by Corollary 10 and Proposition 13 that E(Q n ) < max{2E(Q n ) − 2, E(Q n−1 ) + 1}, which gives the necessary upper b ounds for 8  n  11. We now show that E ′ (Q n ) is at least the value given in the table, which will complete the proof. In dimension 8, we can use Lemma 15 on the jagged set of size 4 in dimen- sion 4 {0001, 0011, 1110, 1111} to obtain the following jagged minimal percolating set in Q 8 : {00010001, 001 1000 1, 11100001, 11110001, 00000110, 00001110, 11001010, 11001110}. In dimension 9, we can simply extend our jagged minimal percolating set in dimen- sion 8 to a jagged minimal percolating set in dimension 9 by embedding Q 8 into Q 9 as ∗ ∗∗∗∗∗∗∗1 and adding the vertex 001111110 to obtain {000100011, 001100011, 111000011, 111100011, 000001101, 000011101, 110010101, 110011101, 001111110}. In dimensions 10 and 11, we can directly apply Lemma 15 to the minimal percolating sets of sizes 5 and 6 in dimensions 6, and 7 resp ectively as given by Lemma 16 to obtain the desired result. 5 Variations In this section we outline some partial results on generalizations of the original question, namely, grids and augmented hypercubes. We hope that this section leads to future study. First, we find an upp er bound on E([n] d , 2). Suppose we have an arbitrary grid the electronic journal of combinatorics 17 (2010), #R80 9 P n 1 × · · · × P n d with A a minimal percolat ing set in t he grid. Then this grid will have many different subcubes in it, of many different dimensions. We find an upper bound on the number of elements of A contained in any subcube, in terms of the dimension d of the subcube. Definition 20. Let G(d) be the maximum over all possible grids, all possible d-dimen- sional subcubes of the grid, and all minimal percolating sets A in the grid of the number of elements of A contained in the subcube. We know G(d) is obviously finite because each cube has only finitely many vertices. We prove an upper bound on G(d). The proof relies heavily on the idea of viewing percolation as combining nearby subcubes, and it looks at the ways tha t the last two cubes in the process can be combined. We set G(d) = 0 for d < 0. We have an obvious analogue of Lemma 8, whose proof is nearly identical, so we omit it. Lemma 21. We have G(d)  G(d − 1). Proposition 22. For d > 1, G(d)  max{G(d − 1) + 1, 2G(d − 3)}. Proof. Let H be a grid, Q ⊂ H a fixed d-dimensional hypercube and A ⊂ H a minimal percolating set with |A∩Q| = G(d). Then since d  0, |A∩Q| > 0. Since A percolates, we know that the final term A s in any execution path will contain a set containing Q, whereas the first term will not. Thus, we can find a k such that A k contains a set containing Q, but A k−1 does not. Hence, the term A k−1 will contain some nonzero number of subgrids which intersect Q. We wish to consider the intersections of these subgrids with Q. Let C 1 , · · · , C j be the set of non-empty intersections of sets in A k−1 with Q, and let C 1 be the largest cube. Note that the C i are all subcubes of Q. Among all possible execution paths, select one with C 1 as large as possible. Among all execution paths with C 1 as large as possible, select one with k as small as possible. Now, if j = 1, then we know that |A ∩ Q| = |C 1 |  G(d − 1), so we are done. Now suppose j > 2. Then at least one of the cubes C i is not necessary to infect Q, since each step in the execution path involves combining only two cubes at a time. This contradicts minimality of k, since otherwise we could reduce k by not performing any of the steps that lead to forming the cubes not used in infecting Q. Thus, we may assume that j = 2 and the two subcubes C 1 and C 2 together infect Q d . By choice of C 1 , dim C 1  dim C 2 . By minimality of A, A ∩ Q ⊂ C 1 ∪ C 2 , and dim C 1  d − 1. We divide into cases depending on dim C 1 . Case 1 dim C 1 = d − 1. Then if (C 2 ∩ A) \ C 1 is empty, we have |A ∩ Q|  G(d − 1) by induction, and we are done. Thus, suppo se there is an x ∈ (C 2 ∩ A) \ C 1 . Then the cube spanned by x and C 1 is a ll of Q. By minimality of k, we know that C 2 = {x}, since otherwise we could find an execution path with smaller k by first combining cubes to infect C 1 and then combining C 1 with x. Thus, |A ∩ Q|  G(d − 1) + 1. Case 2 dim C 1 = d − 2. As before, we are done if (C 2 ∩ A) \ C 1 is empty, so as before let x be a vertex of (C 2 ∩ A) \ C 1 . Then x cannot have distance 1 from C 1 , as we could then find an execution path which would make C 1 larger, namely the path in the electronic journal of combinatorics 17 (2010), #R80 10 [...]... used to model the topological structure of a large-scale parallel processing system For r = 2, a simple inductive argument shows that any minimal percolating set has size 2, since any minimal percolating set must eventually infect two adjacent vertices in a sub-AQn−1 , which will percolate However, using the “wasted” edge-counting technique from [13], we see that E(AQ6 , 7) 14, so percolation is nontrivial... 125(2):195–224, 2003 [11] Robert Morris Minimal percolating sets in bootstrap percolation Electron J Combin., 16(1):Research Paper 2, 20, 2009 [12] Gabor Pete Disease processes and bootstrap percolation Thesis for diploma at the Bolyai Institute, Jzsef Attila University, Szeged, 1997 [13] Eric Riedl Largest and smallest minimal percolating sets in trees Preprint [14] R H Schonmann On the behaviour... (e.g in [11]) it would be very interesting to generalize our hypercube results to r > 2 Question 25 Find m(Qn , r) and E(Qn , r) for r 3 Exploring graphs similar to the hypercube is also likely to be interesting Question 26 Find m(G, r) and E(G, r) for variations of the hypercube such as the augmented hypercube and the twisted hypercube the electronic journal of combinatorics 17 (2010), #R80 11 Finally,... the electronic journal of combinatorics 17 (2010), #R80 11 Finally, it would be interesting to improve our understanding of grids Based on d Corollary 23, we know that E([n] ,2) → 0 as d → ∞, independently of n However, it d n would be interesting to study the question for fixed d, perhaps following Morris [11] or sharpening his result to find precise asymptotics for grids Additionally, given the bounds...which we first combine cubes to create C1 , then combine C1 with the single vertex x Thus, x has distance 2 from C1 As in case 1, we have by minimality of k that C2 = {x}, and so |A ∩ Q| G(d − 2) + 1 Case 3 dim C1 d − 3 Then by definition, |A ∩ C1 | and |A ∩ C2 | are each at most G(d − 3) Hence, |A ∩ Q| 2G(d − 3) in this case Since G(0) = 1, G(1) = 2, G(2) = 2, and G(3) =... 3, we can use the above result to get an upper bound on E([n]d , 2), since G(d) bounds how many vertices of a minimal percolating set of [n]d lie in any subcube of [n]d Corollary 23 For d j 3 1, we have G(d) Corollary 24 We have E([n]d , 2) n d 2 2+ j 3 2 ⌊ d−1 ⌋ 3 where d ≡ j mod 3, 1 G(d) Proof Simply partition the grid into hypercubes and apply the previous corollary Now we consider the augmented... Balogh, B Bollob´s, H Duminil-Copin, and R Morris The sharp metastability a threshold for r-neighbor bootstrap percolation In preparation [5] J Balogh, B Bollob´s, and R Morris Bootstrap percolation in three dimensions a Ann Probab., 37:1329–1380, 2009 [6] J Balogh, Y Peres, and G Pete Bootstrap percolation on in nite trees and nonamenable groups Combinatorics, Probability and Computing, 15:715–730, 2006... many related problems left to consider, and as we gain more understanding of the percolation process through studying this extremal problem, it is reasonable to hope that we will be able to make more progress on the original probabilistic question We would like to know m(G, r) and E(G, r) for any finite graph At the moment however, it seems that finding a general formula or algorithm is too ambitious... natural to wish to investigate the behavior of α(n, d) = E([n]d ,2) nd 1/d Question 27 Find precise asymptotics for E([n]d , r) Acknowledgements I would like to thank Joe Gallian for bringing the problem to my attention and for all his help I would like to thank Nathan Kaplan and Nathan Pflueger for all of their help with this paper I would also like to thank Sasha Ovestsky Fradkin for her advice on... Combinatorics, Probability and Computing, 15:715–730, 2006 [7] R Cerf and E Cirillo Finite size scaling in three-dimensional bootstrap percolation Ann Probab., 27(4):1837–1850, 1999 [8] R Cerf and F Manzo The threshold regime of finite volume bootstrap percolation Stochastic Process Appl., 101(1):69–82, 2002 the electronic journal of combinatorics 17 (2010), #R80 12 [9] J Chalupa, P L Leath, and G R Reich Bootstrap . for constructing large minimal percolating sets in high dimensions. Thus, we define a type of minimal percolating set which is suited to constructing large minimal percolating sets in high dimensions Suppose we have a minimal percolating set A in Q n−4 which is jagged. Then we construct a jagged minimal percolating set B of size 2|A| in Q n . We build up our minimal percolating set B in two halves,. is a minimal percolating set in Q n with A = B ∪ C the disjoint union of two minimal percolating sets in subcubes of dimension n − 4. Then neither B nor C is isomorphic to the constructioin in